PUBLIC GOODS Snyder and Nicholson Copyright 2008

Public Goods • Public goods are nonrival – the use of the good does not prevent others from using it e. g. knowledge • Pure Public goods are nonexclusive – once they are produced, they provide benefits to an entire group – it is impossible to restrict these benefits to the specific groups of individuals who pay for them

Attributes of Public Goods • A good is nonrival if consumption of additional units of the good involves zero social marginal costs of production

Attributes of Public Goods • A good is exclusive if it is relatively easy to exclude individuals from benefiting from the good once it is produced • A good is nonexclusive if it is impossible, or very costly, to exclude individuals from benefiting from the good

Attributes of Public Goods • Some examples of these types of goods include:

Public Good • A good is a pure public good if, once produced, no one can be excluded from benefiting from its availability and if the good is nonrival -- the marginal cost of an additional consumer is zero

Public Goods and Resource Allocation • We will use a simple general equilibrium model with two individuals (A and B) • There are only two goods – good y is an ordinary private good • each person begins with an allocation (y. A and y. B) – good x is a public good that is produced using y x = f(ys. A + ys. B)

Public Goods and Resource Allocation • Resulting utilities for these individuals are UA[x, (y. A - ys. A)] UB[x, (y. B - ys. B)] • The level of x enters identically into each person’s utility curve – it is nonexclusive and nonrival • each person’s consumption is unrelated to what he contributes to production • each consumes the total amount produced

Public Goods and Resource Allocation • The necessary conditions for efficient resource allocation consist of choosing the levels of ys. A and ys. B that maximize one person’s (A’s) utility for any given level of the other’s (B’s) utility • The Lagrangian expression is L = UA(x, y. A - ys. A) + [UB(x, y. B - ys. B) - K]

Public Goods and Resource Allocation • The first-order conditions for a maximum are L/ ys. A = U 1 Af’ - U 2 A + U 1 Bf’ = 0 L/ ys. B = U 1 Af’ - U 2 B + U 1 Bf’ = 0 • Comparing the two equations, we find U 2 B = U 2 A

Public Goods and Resource Allocation • We can now derive the optimality condition for the production of x • From the initial first-order condition we know that U 1 A/U 2 A + U 1 B/ U 2 B = 1/f’ MRSA + MRSB = 1/f’ • The MRS must reflect all consumers because all will get the same benefits

Failure of a Competitive Market • Production of x and y in competitive markets will fail to achieve this allocation – with perfectly competitive prices px and py, each individual will equate his MRS to px/py – the producer will also set 1/f’ equal to px/py to maximize profits – the price ratio px/py will be too low • it would provide too little incentive to produce x

Failure of a Competitive Market • For public goods, the value of producing one more unit is the sum of each consumer’s valuation of that output – individual demand curves should be added vertically rather than horizontally • Thus, the usual market demand curve will not reflect the full marginal valuation

Inefficiency of a Nash Equilibrium • Suppose that individual A is thinking about contributing ys. A of his initial y. A endowment to the production of x • The utility maximization problem for A is then choose ys. A to maximize UA[f(ys. A + ys. B), y. A - ys. A]

Inefficiency of a Nash Equilibrium • The first-order condition for a maximum is U 1 Af’ - U 2 A = 0 U 1 A/U 2 A = MRSA = 1/f’ • Because a similar argument can be applied to B, the efficiency condition will fail to be achieved – each person considers only his own benefit

The Roommates’ Dilemma • Suppose two roommates with identical preferences derive utility from the number of paintings hung on their walls (x) and the number of chocolate bars they eat (y) with a utility function of Ui(x, yi) = x 1/3 yi 2/3 (for i=1, 2) • Assume each roommate has $300 to spend and that px = $100 and py = $0. 20

The Roommates’ Dilemma • We know from our earlier analysis of Cobb -Douglas utility functions that if each individual lived alone, he would spend 1/3 of his income on paintings (x = 1) and 2/3 on chocolate bars (y = 1, 000) • When the roommates live together, each must consider what the other will do – if each assumed the other would buy paintings, x = 0 and utility = 0

The Roommates’ Dilemma • If person 1 believes that person 2 will not buy any paintings, he could choose to purchase one and receive utility of U 1(x, y 1) = 11/3(1, 000)2/3 = 100 while person 2’s utility will be U 2(x, y 2) = 11/3(1, 500)2/3 = 131 • Person 2 has gained from his free-riding position

The Roommates’ Dilemma • We can show that this solution is inefficient by calculating each person’s MRS • At the allocations described, MRS 1 = 1, 000/2 = 500 MRS 2 = 1, 500/2 = 750

The Roommates’ Dilemma • Since MRS 1 + MRS 2 = 1, 250, the roommates would be willing to sacrifice 1, 250 chocolate bars to have one additional painting – an additional painting would only cost them 500 chocolate bars – too few paintings are bought

The Roommates’ Dilemma • To calculate the efficient level of x, we must set the sum of each person’s MRS equal to the price ratio • This means that y 1 + y 2 = 1, 000 x

The Roommates’ Dilemma • Substituting into the sum of budget constraints, we get 0. 20(y 1 + y 2) + 100 x = 600 x=2 y 1 + y 2 = 2, 000 • The allocation of the cost of the paintings depends on how each roommate plays the strategic financing game

The Roommates’ Dilemma • If each person buy 1 painting Person 1’s utility is U 1(x, y 1) = 21/3(1, 000)2/3 126 Person 2’s utility is U 2(x, y 2) = 21/3(1, 000)2/3 126

The Roommates’ Dilemma Person 2 Buy Not Buy 126, 126 100, 131 Person 1 Not Buy 131, 100 0, 0

Lindahl Pricing of Public Goods • Swedish economist E. Lindahl suggested that individuals might be willing to be taxed for public goods if they knew that others were being taxed – Lindahl assumed that each individual would be presented by the government with the proportion of a public good’s cost he was expected to pay and then reply with the level of public good he would prefer

Lindahl Pricing of Public Goods • Suppose that individual A would be quoted a specific percentage ( A) and asked the level of a public good (x) he would want given the knowledge that this fraction of total cost would have to be paid • The person would choose the level of x which maximizes utility = UA[x, y. A*- Af -1(x)]

Lindahl Pricing of Public Goods • The first-order condition is given by U 1 A - AU 2 B(1/f’)=0 MRSA = A/f’ • Faced by the same choice, individual B would opt for the level of x which satisfies MRSB = B/f’

Lindahl Pricing of Public Goods • An equilibrium would occur when A+ B = 1 – the level of public goods expenditure favored by the two individuals precisely generates enough tax contributions to pay for it MRSA + MRSB = ( A + B)/f’ = 1/f’

Shortcomings of the Lindahl Solution • The incentive to be a free rider is very strong – this makes it difficult to envision how the information necessary to compute equilibrium Lindahl shares might be computed • individuals have a clear incentive to understate their true preferences

Important Points to Note: • Externalities may cause a misallocation of resources because of a divergence between private and social marginal cost – traditional solutions to this divergence includes mergers among the affected parties and adoption of suitable Pigouvian taxes or subsidies

Voting • Voting is used as a social decision process in many institutions – direct voting is used in many cases from statewide referenda to smaller groups and clubs – in other cases, societies have found it more convenient to use a representative form of government

Majority Rule • Throughout our discussion of voting, we will assume that decisions will be made by majority rule

The Paradox of Voting • In the 1780 s, social theorist M. de Condorcet noted that majority rule voting systems may not arrive at an equilibrium – instead, they may cycle among alternative options

The Paradox of Voting • Suppose there are three voters (Smith, Jones, and Fudd) choosing among three policy options – we can assume that these policy options represent three levels of spending on a particular public good [(A) low, (B) medium, and (C) high] – Condorcet’s paradox would arise even without this ordering

The Paradox of Voting • Preferences among the three policy options for the three voters are:

The Paradox of Voting • Consider a vote between A and B – A would win • In a vote between A and C – C would win • In a vote between B and C – B would win • No equilibrium will ever be reached

Single-Peaked Preferences • Equilibrium voting outcomes always occur in cases where the issue being voted upon is one-dimensional and where voter preferences are “singlepeaked”

Single-Peaked Preferences We can show each voters preferences in terms of utility levels Utility A Fudd Jones B C For Smith and Jones, preferences are singlepeaked Fudd’s preferences have two local maxima Smith Quantity of public good

Single-Peaked Preferences If Fudd had alternative preferences with a single peak, there would be no paradox Utility A Fudd Jones Option B will be chosen because it will defeat both A and C by votes 2 to 1 B C Smith Quantity of public good

The Median Voter Theorem • With the altered preferences of Fudd, B will be chosen because it is the preferred choice of the median voter (Jones) – Jones’s preferences are between the preferences of Smith and the revised preferences of Fudd

The Median Voter Theorem • If choices are unidimensional and preferences are single-peaked, majority rule will result in the selection of the project that is most favored by the median voter – that voter’s preferences will determine what public choices are made

A Simple Political Model • Suppose a community is characterized by a large number of voters (n) each with income of yi • The utility of each voter depends on his consumption of a private good (ci) and of a public good (g) according to utility of person i = Ui = ci + f(g) where fg > 0 and fgg < 0

A Simple Political Model • Each voter must pay taxes to finance g • Taxes are proportional to income and are imposed at a rate of t • Each person’s budget constraint is ci = (1 -t)yi • The government also faces a budget constraint

A Simple Political Model • Given these constraints, the utility function of individual i is Ui(g) = [y. A - (g/n)]yi /y. A + f(g) • Utility maximization occurs when d. Ui /dg = -yi /(ny. A) + fg(g) = 0 g = fg-1[yi /(ny. A)] • Desired spending on g is inversely related to income

A Simple Political Model • If G is determined through majority rule, its level will be that level favored by the median voter – since voters’ preferences are determined solely by income, g will be set at the level preferred by the voter with the median level of income (ym) g* = fg-1[ym/(ny. A)] = fg-1[(1/n)(ym/y. A)]

A Simple Political Model • Under a utilitarian social welfare criterion, g would be chosen so as to maximize the sum of utilities: • The optimal choice for g then is g* = fg-1(1/n) = fg-1[(1/n)(y. A/y. A)] – the level of g favored by the voter with average income

Voting for Redistributive Taxation • Suppose voters are considering a lumpsum transfer to be paid to every person and financed through proportional taxation • If we denote the per-person transfer b, each individual’s utility is now given by Ui = ci + b

Voting for Redistributive Taxation • The government’s budget constraint is nb = tny. A b = ty. A • For a voter with yi > y. A, utility is maximized by choosing b = 0 • Any voter with yi < y. A will choose t = 1 and b = y. A – would fully equalize incomes

Voting for Redistributive Taxation • Note that a 100 percent tax rate would lower average income • Assume that each individual’s income has two components, one responsive to tax rates [yi (t)] and one not responsive (ni) – also assume that the average of ni is zero, but its distribution is skewed right so nm < 0

Voting for Redistributive Taxation • Now, utility is given by Ui = (1 -t)[yi (t) + ni] + b • The individual’s first-order condition for a maximum in his choice of t and g is now d. Ui /dt = -ni + t(dy. A/dt) = 0 ti = ni /(dy. A/dt) • Under majority rule, the equilibrium condition will be t* = nm /(dy. A/dt)

The Groves Mechanism • Suppose there are n individuals in a group – each has a private and unobservable valuation (ui) for a proposed taxationexpenditure project

The Groves Mechanism • The government states that, if undertaken, the project will provide each person with a transfer given by • If the project is not undertaken, no transfers are made

The Groves Mechanism • The problem for voter i is to choose his or her reported net valuation so as to maximize utility

The Groves Mechanism • Each person will wish the project to be undertaken if it raises utility • This means • A utility-maximizing strategy is to set vi = ui

The Clarke Mechanism • This mechanism also envisions asking individuals about their net valuation of a public project – focuses on “pivotal voters” • those whose valuations can change the overall evaluation from positive to negative of vice versa

The Clarke Mechanism • For these pivotal voters, the Clarke mechanism incorporates a Pigouvian-like tax (or transfer) to encourage truth telling – for all other voters, there are no special transfers

Important Points to Note: • Public goods provide benefits to individuals on a nonexclusive basis no one can be prevented from consuming such goods – such goods are usually nonrival in that the marginal cost of serving another user is zero

Important Points to Note: • Private markets will tend to underallocate resources to public goods because no single buyer can appropriate all of the benefits that such goods provide

Important Points to Note: • A Lindahl optimal tax-sharing scheme can result in an efficient allocation of resources to the production of public goods – computing these tax shares requires substantial information that individuals have incentives to hide

Important Points to Note: • Majority rule voting may not lead to an efficient allocation of resources to public goods – the median voter theorem provides a useful way of modeling the outcomes from majority rule in certain situations

Important Points to Note: • Several truth-revealing voting mechanisms have been developed – whether these are robust to the special assumptions made or capable of practical application remain unresolved questions